LMFDB - Embedded newform 669.2.a.h.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [669,2,Mod(1,669)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("669.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(669, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 669 = 3 \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	669.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,2,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$5.34199189522$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 8x^{5} + 15x^{4} + 15x^{3} - 26x^{2} + 2x + 1 $ x^7 - 2x^6 - 8x^5 + 15x^4 + 15x^3 - 26x^2 + 2x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.156720$ of defining polynomial
Character	$\chi$	$=$	669.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.156720 q^{2} -1.00000 q^{3} -1.97544 q^{4} +2.84044 q^{5} +0.156720 q^{6} +2.12887 q^{7} +0.623032 q^{8} +1.00000 q^{9} -0.445154 q^{10} -1.54035 q^{11} +1.97544 q^{12} -2.19124 q^{13} -0.333637 q^{14} -2.84044 q^{15} +3.85324 q^{16} -2.31004 q^{17} -0.156720 q^{18} +3.87780 q^{19} -5.61111 q^{20} -2.12887 q^{21} +0.241405 q^{22} +6.13143 q^{23} -0.623032 q^{24} +3.06807 q^{25} +0.343412 q^{26} -1.00000 q^{27} -4.20545 q^{28} -1.33716 q^{29} +0.445154 q^{30} +5.18277 q^{31} -1.84995 q^{32} +1.54035 q^{33} +0.362030 q^{34} +6.04692 q^{35} -1.97544 q^{36} +4.11852 q^{37} -0.607730 q^{38} +2.19124 q^{39} +1.76968 q^{40} +8.48963 q^{41} +0.333637 q^{42} -2.54730 q^{43} +3.04287 q^{44} +2.84044 q^{45} -0.960921 q^{46} +5.81331 q^{47} -3.85324 q^{48} -2.46792 q^{49} -0.480830 q^{50} +2.31004 q^{51} +4.32866 q^{52} +14.1513 q^{53} +0.156720 q^{54} -4.37528 q^{55} +1.32635 q^{56} -3.87780 q^{57} +0.209561 q^{58} +3.25954 q^{59} +5.61111 q^{60} +9.61946 q^{61} -0.812246 q^{62} +2.12887 q^{63} -7.41655 q^{64} -6.22407 q^{65} -0.241405 q^{66} -0.700772 q^{67} +4.56334 q^{68} -6.13143 q^{69} -0.947675 q^{70} -1.01618 q^{71} +0.623032 q^{72} -11.7272 q^{73} -0.645456 q^{74} -3.06807 q^{75} -7.66035 q^{76} -3.27921 q^{77} -0.343412 q^{78} +2.71486 q^{79} +10.9449 q^{80} +1.00000 q^{81} -1.33050 q^{82} +11.2303 q^{83} +4.20545 q^{84} -6.56151 q^{85} +0.399214 q^{86} +1.33716 q^{87} -0.959690 q^{88} -9.83376 q^{89} -0.445154 q^{90} -4.66486 q^{91} -12.1123 q^{92} -5.18277 q^{93} -0.911064 q^{94} +11.0146 q^{95} +1.84995 q^{96} -8.34884 q^{97} +0.386773 q^{98} -1.54035 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{2} - 7 q^{3} + 6 q^{4} + 3 q^{5} - 2 q^{6} - 2 q^{7} + 3 q^{8} + 7 q^{9} + 15 q^{11} - 6 q^{12} - 2 q^{13} + 4 q^{14} - 3 q^{15} + 5 q^{17} + 2 q^{18} + 20 q^{19} + 8 q^{20} + 2 q^{21} + 11 q^{22}+ \cdots + 15 q^{99}+O(q^{100}) $ 7 * q + 2 * q^2 - 7 * q^3 + 6 * q^4 + 3 * q^5 - 2 * q^6 - 2 * q^7 + 3 * q^8 + 7 * q^9 + 15 * q^11 - 6 * q^12 - 2 * q^13 + 4 * q^14 - 3 * q^15 + 5 * q^17 + 2 * q^18 + 20 * q^19 + 8 * q^20 + 2 * q^21 + 11 * q^22 + 9 * q^23 - 3 * q^24 - 4 * q^25 + 19 * q^26 - 7 * q^27 - 8 * q^28 + 9 * q^29 + 10 * q^31 + 8 * q^32 - 15 * q^33 + q^34 + 12 * q^35 + 6 * q^36 - 16 * q^37 + 25 * q^38 + 2 * q^39 + 17 * q^40 + 16 * q^41 - 4 * q^42 - 5 * q^43 + 26 * q^44 + 3 * q^45 + 2 * q^46 + 21 * q^47 + 3 * q^49 + 6 * q^50 - 5 * q^51 - 9 * q^52 + 5 * q^53 - 2 * q^54 + 7 * q^55 + 14 * q^56 - 20 * q^57 + 2 * q^58 + 26 * q^59 - 8 * q^60 + 12 * q^61 + q^62 - 2 * q^63 - 41 * q^64 - 4 * q^65 - 11 * q^66 - 3 * q^67 + 8 * q^68 - 9 * q^69 - 34 * q^70 + 21 * q^71 + 3 * q^72 - 23 * q^73 - 10 * q^74 + 4 * q^75 + 35 * q^76 - 16 * q^77 - 19 * q^78 + 13 * q^79 - 4 * q^80 + 7 * q^81 - 15 * q^82 + 26 * q^83 + 8 * q^84 - 25 * q^85 - 44 * q^86 - 9 * q^87 + 15 * q^88 + 20 * q^89 + 30 * q^91 - 5 * q^92 - 10 * q^93 + 5 * q^94 + 10 * q^95 - 8 * q^96 - 9 * q^97 - 32 * q^98 + 15 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.156720	−0.110818	−0.0554090	−	0.998464i	$-0.517646\pi$
$2$	−0.156720	−0.110818	−0.0554090	+	0.998464i	$0.517646\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.97544	−0.987719
$5$	2.84044	1.27028	0.635141	−	0.772396i	$-0.280942\pi$
$5$	2.84044	1.27028	0.635141	+	0.772396i	$0.280942\pi$
$6$	0.156720	0.0639808
$7$	2.12887	0.804637	0.402318	−	0.915500i	$-0.368204\pi$
$7$	2.12887	0.804637	0.402318	+	0.915500i	$0.368204\pi$
$8$	0.623032	0.220275
$9$	1.00000	0.333333
$10$	−0.445154	−0.140770
$11$	−1.54035	−0.464434	−0.232217	−	0.972664i	$-0.574598\pi$
$11$	−1.54035	−0.464434	−0.232217	+	0.972664i	$0.574598\pi$
$12$	1.97544	0.570260
$13$	−2.19124	−0.607740	−0.303870	−	0.952714i	$-0.598279\pi$
$13$	−2.19124	−0.607740	−0.303870	+	0.952714i	$0.598279\pi$
$14$	−0.333637	−0.0891683
$15$	−2.84044	−0.733397
$16$	3.85324	0.963309
$17$	−2.31004	−0.560266	−0.280133	−	0.959961i	$-0.590379\pi$
$17$	−2.31004	−0.560266	−0.280133	+	0.959961i	$0.590379\pi$
$18$	−0.156720	−0.0369394
$19$	3.87780	0.889628	0.444814	−	0.895623i	$-0.353270\pi$
$19$	3.87780	0.889628	0.444814	+	0.895623i	$0.353270\pi$
$20$	−5.61111	−1.25468
$21$	−2.12887	−0.464557
$22$	0.241405	0.0514677
$23$	6.13143	1.27849	0.639246	−	0.769002i	$-0.279246\pi$
$23$	6.13143	1.27849	0.639246	+	0.769002i	$0.279246\pi$
$24$	−0.623032	−0.127176
$25$	3.06807	0.613615
$26$	0.343412	0.0673486
$27$	−1.00000	−0.192450
$28$	−4.20545	−0.794755
$29$	−1.33716	−0.248305	−0.124153	−	0.992263i	$-0.539621\pi$
$29$	−1.33716	−0.248305	−0.124153	+	0.992263i	$0.539621\pi$
$30$	0.445154	0.0812737
$31$	5.18277	0.930853	0.465427	−	0.885086i	$-0.345901\pi$
$31$	5.18277	0.930853	0.465427	+	0.885086i	$0.345901\pi$
$32$	−1.84995	−0.327027
$33$	1.54035	0.268141
$34$	0.362030	0.0620876
$35$	6.04692	1.02212
$36$	−1.97544	−0.329240
$37$	4.11852	0.677081	0.338540	−	0.940952i	$-0.390067\pi$
$37$	4.11852	0.677081	0.338540	+	0.940952i	$0.390067\pi$
$38$	−0.607730	−0.0985868
$39$	2.19124	0.350879
$40$	1.76968	0.279812
$41$	8.48963	1.32586	0.662929	−	0.748683i	$-0.269313\pi$
$41$	8.48963	1.32586	0.662929	+	0.748683i	$0.269313\pi$
$42$	0.333637	0.0514813
$43$	−2.54730	−0.388459	−0.194230	−	0.980956i	$-0.562221\pi$
$43$	−2.54730	−0.388459	−0.194230	+	0.980956i	$0.562221\pi$
$44$	3.04287	0.458731
$45$	2.84044	0.423427
$46$	−0.960921	−0.141680
$47$	5.81331	0.847958	0.423979	−	0.905672i	$-0.360633\pi$
$47$	5.81331	0.847958	0.423979	+	0.905672i	$0.360633\pi$
$48$	−3.85324	−0.556167
$49$	−2.46792	−0.352559
$50$	−0.480830	−0.0679996
$51$	2.31004	0.323470
$52$	4.32866	0.600277
$53$	14.1513	1.94384	0.971918	−	0.235318i	$-0.0756131\pi$
$53$	14.1513	1.94384	0.971918	+	0.235318i	$0.0756131\pi$
$54$	0.156720	0.0213269
$55$	−4.37528	−0.589962
$56$	1.32635	0.177242
$57$	−3.87780	−0.513627
$58$	0.209561	0.0275167
$59$	3.25954	0.424356	0.212178	−	0.977231i	$-0.431944\pi$
$59$	3.25954	0.424356	0.212178	+	0.977231i	$0.431944\pi$
$60$	5.61111	0.724391
$61$	9.61946	1.23165	0.615823	−	0.787885i	$-0.288824\pi$
$61$	9.61946	1.23165	0.615823	+	0.787885i	$0.288824\pi$
$62$	−0.812246	−0.103155
$63$	2.12887	0.268212
$64$	−7.41655	−0.927068
$65$	−6.22407	−0.772001
$66$	−0.241405	−0.0297149
$67$	−0.700772	−0.0856129	−0.0428065	−	0.999083i	$-0.513630\pi$
$67$	−0.700772	−0.0856129	−0.0428065	+	0.999083i	$0.513630\pi$
$68$	4.56334	0.553386
$69$	−6.13143	−0.738138
$70$	−0.947675	−0.113269
$71$	−1.01618	−0.120598	−0.0602990	−	0.998180i	$-0.519205\pi$
$71$	−1.01618	−0.120598	−0.0602990	+	0.998180i	$0.519205\pi$
$72$	0.623032	0.0734251
$73$	−11.7272	−1.37257	−0.686284	−	0.727334i	$-0.740759\pi$
$73$	−11.7272	−1.37257	−0.686284	+	0.727334i	$0.740759\pi$
$74$	−0.645456	−0.0750328
$75$	−3.06807	−0.354271
$76$	−7.66035	−0.878702
$77$	−3.27921	−0.373701
$78$	−0.343412	−0.0388837
$79$	2.71486	0.305445	0.152723	−	0.988269i	$-0.451196\pi$
$79$	2.71486	0.305445	0.152723	+	0.988269i	$0.451196\pi$
$80$	10.9449	1.22367
$81$	1.00000	0.111111
$82$	−1.33050	−0.146929
$83$	11.2303	1.23269	0.616344	−	0.787477i	$-0.288613\pi$
$83$	11.2303	1.23269	0.616344	+	0.787477i	$0.288613\pi$
$84$	4.20545	0.458852
$85$	−6.56151	−0.711696
$86$	0.399214	0.0430483
$87$	1.33716	0.143359
$88$	−0.959690	−0.102303
$89$	−9.83376	−1.04238	−0.521188	−	0.853442i	$-0.674511\pi$
$89$	−9.83376	−1.04238	−0.521188	+	0.853442i	$0.674511\pi$
$90$	−0.445154	−0.0469234
$91$	−4.66486	−0.489010
$92$	−12.1123	−1.26279
$93$	−5.18277	−0.537428
$94$	−0.911064	−0.0939691
$95$	11.0146	1.13008
$96$	1.84995	0.188809
$97$	−8.34884	−0.847696	−0.423848	−	0.905733i	$-0.639321\pi$
$97$	−8.34884	−0.847696	−0.423848	+	0.905733i	$0.639321\pi$
$98$	0.386773	0.0390700
$99$	−1.54035	−0.154811
$100$	−6.06079	−0.606079
$101$	−8.46921	−0.842718	−0.421359	−	0.906894i	$-0.638447\pi$
$101$	−8.46921	−0.842718	−0.421359	+	0.906894i	$0.638447\pi$
$102$	−0.362030	−0.0358463
$103$	−14.5237	−1.43106	−0.715532	−	0.698580i	$-0.753816\pi$
$103$	−14.5237	−1.43106	−0.715532	+	0.698580i	$0.753816\pi$
$104$	−1.36521	−0.133870
$105$	−6.04692	−0.590119
$106$	−2.21780	−0.215412
$107$	9.79683	0.947095	0.473548	−	0.880768i	$-0.342973\pi$
$107$	9.79683	0.947095	0.473548	+	0.880768i	$0.342973\pi$
$108$	1.97544	0.190087
$109$	−10.4368	−0.999665	−0.499833	−	0.866122i	$-0.666605\pi$
$109$	−10.4368	−0.999665	−0.499833	+	0.866122i	$0.666605\pi$
$110$	0.685695	0.0653785
$111$	−4.11852	−0.390913
$112$	8.20303	0.775114
$113$	−12.0988	−1.13815	−0.569077	−	0.822284i	$-0.692700\pi$
$113$	−12.0988	−1.13815	−0.569077	+	0.822284i	$0.692700\pi$
$114$	0.607730	0.0569191
$115$	17.4159	1.62405
$116$	2.64149	0.245256
$117$	−2.19124	−0.202580
$118$	−0.510836	−0.0470263
$119$	−4.91777	−0.450811
$120$	−1.76968	−0.161549
$121$	−8.62731	−0.784301
$122$	−1.50757	−0.136489
$123$	−8.48963	−0.765484
$124$	−10.2382	−0.919422
$125$	−5.48751	−0.490818
$126$	−0.333637	−0.0297228
$127$	−1.34751	−0.119572	−0.0597861	−	0.998211i	$-0.519042\pi$
$127$	−1.34751	−0.119572	−0.0597861	+	0.998211i	$0.519042\pi$
$128$	4.86222	0.429763
$129$	2.54730	0.224277
$130$	0.975439	0.0855516
$131$	11.5131	1.00591	0.502953	−	0.864314i	$-0.332247\pi$
$131$	11.5131	1.00591	0.502953	+	0.864314i	$0.332247\pi$
$132$	−3.04287	−0.264848
$133$	8.25532	0.715827
$134$	0.109825	0.00948746
$135$	−2.84044	−0.244466
$136$	−1.43923	−0.123413
$137$	5.35345	0.457376	0.228688	−	0.973500i	$-0.426556\pi$
$137$	5.35345	0.457376	0.228688	+	0.973500i	$0.426556\pi$
$138$	0.960921	0.0817990
$139$	1.04031	0.0882379	0.0441190	−	0.999026i	$-0.485952\pi$
$139$	1.04031	0.0882379	0.0441190	+	0.999026i	$0.485952\pi$
$140$	−11.9453	−1.00956
$141$	−5.81331	−0.489569
$142$	0.159256	0.0133644
$143$	3.37528	0.282255
$144$	3.85324	0.321103
$145$	−3.79813	−0.315417
$146$	1.83790	0.152105
$147$	2.46792	0.203550
$148$	−8.13589	−0.668766
$149$	0.386355	0.0316515	0.0158257	−	0.999875i	$-0.494962\pi$
$149$	0.386355	0.0316515	0.0158257	+	0.999875i	$0.494962\pi$
$150$	0.480830	0.0392596
$151$	10.8108	0.879769	0.439885	−	0.898054i	$-0.355019\pi$
$151$	10.8108	0.879769	0.439885	+	0.898054i	$0.355019\pi$
$152$	2.41599	0.195963
$153$	−2.31004	−0.186755
$154$	0.513920	0.0414128
$155$	14.7213	1.18245
$156$	−4.32866	−0.346570
$157$	−9.32975	−0.744595	−0.372297	−	0.928113i	$-0.621430\pi$
$157$	−9.32975	−0.744595	−0.372297	+	0.928113i	$0.621430\pi$
$158$	−0.425473	−0.0338488
$159$	−14.1513	−1.12227
$160$	−5.25465	−0.415417
$161$	13.0530	1.02872
$162$	−0.156720	−0.0123131
$163$	−14.0026	−1.09677	−0.548383	−	0.836227i	$-0.684756\pi$
$163$	−14.0026	−1.09677	−0.548383	+	0.836227i	$0.684756\pi$
$164$	−16.7707	−1.30957
$165$	4.37528	0.340615
$166$	−1.76002	−0.136604
$167$	17.6253	1.36389	0.681945	−	0.731403i	$-0.261134\pi$
$167$	17.6253	1.36389	0.681945	+	0.731403i	$0.261134\pi$
$168$	−1.32635	−0.102330
$169$	−8.19848	−0.630652
$170$	1.02832	0.0788688
$171$	3.87780	0.296543
$172$	5.03203	0.383689
$173$	0.820392	0.0623733	0.0311866	−	0.999514i	$-0.490071\pi$
$173$	0.820392	0.0623733	0.0311866	+	0.999514i	$0.490071\pi$
$174$	−0.209561	−0.0158868
$175$	6.53153	0.493737
$176$	−5.93535	−0.447394
$177$	−3.25954	−0.245002
$178$	1.54115	0.115514
$179$	−3.57848	−0.267468	−0.133734	−	0.991017i	$-0.542697\pi$
$179$	−3.57848	−0.267468	−0.133734	+	0.991017i	$0.542697\pi$
$180$	−5.61111	−0.418227
$181$	4.90607	0.364666	0.182333	−	0.983237i	$-0.441635\pi$
$181$	4.90607	0.364666	0.182333	+	0.983237i	$0.441635\pi$
$182$	0.731079	0.0541911
$183$	−9.61946	−0.711091
$184$	3.82008	0.281620
$185$	11.6984	0.860083
$186$	0.812246	0.0595568
$187$	3.55828	0.260207
$188$	−11.4838	−0.837545
$189$	−2.12887	−0.154852
$190$	−1.72622	−0.125233
$191$	8.58557	0.621230	0.310615	−	0.950536i	$-0.399465\pi$
$191$	8.58557	0.621230	0.310615	+	0.950536i	$0.399465\pi$
$192$	7.41655	0.535243
$193$	14.2223	1.02375	0.511873	−	0.859061i	$-0.328952\pi$
$193$	14.2223	1.02375	0.511873	+	0.859061i	$0.328952\pi$
$194$	1.30843	0.0939400
$195$	6.22407	0.445715
$196$	4.87522	0.348230
$197$	−6.37745	−0.454375	−0.227187	−	0.973851i	$-0.572953\pi$
$197$	−6.37745	−0.454375	−0.227187	+	0.973851i	$0.572953\pi$
$198$	0.241405	0.0171559
$199$	−9.87698	−0.700161	−0.350080	−	0.936720i	$-0.613846\pi$
$199$	−9.87698	−0.700161	−0.350080	+	0.936720i	$0.613846\pi$
$200$	1.91151	0.135164
$201$	0.700772	0.0494287
$202$	1.32730	0.0933884
$203$	−2.84665	−0.199795
$204$	−4.56334	−0.319498
$205$	24.1142	1.68421
$206$	2.27616	0.158588
$207$	6.13143	0.426164
$208$	−8.44335	−0.585441
$209$	−5.97318	−0.413173
$210$	0.947675	0.0653958
$211$	2.07499	0.142848	0.0714242	−	0.997446i	$-0.477246\pi$
$211$	2.07499	0.142848	0.0714242	+	0.997446i	$0.477246\pi$
$212$	−27.9551	−1.91997
$213$	1.01618	0.0696272
$214$	−1.53536	−0.104955
$215$	−7.23544	−0.493453
$216$	−0.623032	−0.0423920
$217$	11.0334	0.748999
$218$	1.63566	0.110781
$219$	11.7272	0.792453
$220$	8.64309	0.582717
$221$	5.06184	0.340496
$222$	0.645456	0.0433202
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	−3.93829	−0.263138
$225$	3.06807	0.204538
$226$	1.89612	0.126128
$227$	−0.624446	−0.0414459	−0.0207230	−	0.999785i	$-0.506597\pi$
$227$	−0.624446	−0.0414459	−0.0207230	+	0.999785i	$0.506597\pi$
$228$	7.66035	0.507319
$229$	22.7596	1.50399	0.751997	−	0.659167i	$-0.229091\pi$
$229$	22.7596	1.50399	0.751997	+	0.659167i	$0.229091\pi$
$230$	−2.72943	−0.179974
$231$	3.27921	0.215756
$232$	−0.833097	−0.0546955
$233$	−13.6963	−0.897275	−0.448638	−	0.893714i	$-0.648091\pi$
$233$	−13.6963	−0.897275	−0.448638	+	0.893714i	$0.648091\pi$
$234$	0.343412	0.0224495
$235$	16.5123	1.07715
$236$	−6.43902	−0.419144
$237$	−2.71486	−0.176349
$238$	0.770715	0.0499580
$239$	−5.70450	−0.368993	−0.184497	−	0.982833i	$-0.559066\pi$
$239$	−5.70450	−0.368993	−0.184497	+	0.982833i	$0.559066\pi$
$240$	−10.9449	−0.706488
$241$	−26.0686	−1.67923	−0.839613	−	0.543184i	$-0.817218\pi$
$241$	−26.0686	−1.67923	−0.839613	+	0.543184i	$0.817218\pi$
$242$	1.35208	0.0869147
$243$	−1.00000	−0.0641500
$244$	−19.0026	−1.21652
$245$	−7.00996	−0.447850
$246$	1.33050	0.0848295
$247$	−8.49717	−0.540662
$248$	3.22904	0.205044
$249$	−11.2303	−0.711693
$250$	0.860005	0.0543915
$251$	−13.0234	−0.822032	−0.411016	−	0.911628i	$-0.634826\pi$
$251$	−13.0234	−0.822032	−0.411016	+	0.911628i	$0.634826\pi$
$252$	−4.20545	−0.264918
$253$	−9.44458	−0.593776
$254$	0.211182	0.0132508
$255$	6.56151	0.410898
$256$	14.0711	0.879443
$257$	−11.5247	−0.718893	−0.359446	−	0.933166i	$-0.617034\pi$
$257$	−11.5247	−0.718893	−0.359446	+	0.933166i	$0.617034\pi$
$258$	−0.399214	−0.0248540
$259$	8.76779	0.544804
$260$	12.2953	0.762520
$261$	−1.33716	−0.0827684
$262$	−1.80434	−0.111473
$263$	−17.7196	−1.09264	−0.546319	−	0.837577i	$-0.683971\pi$
$263$	−17.7196	−1.09264	−0.546319	+	0.837577i	$0.683971\pi$
$264$	0.959690	0.0590649
$265$	40.1960	2.46922
$266$	−1.29378	−0.0793266
$267$	9.83376	0.601816
$268$	1.38433	0.0845616
$269$	−15.3018	−0.932966	−0.466483	−	0.884530i	$-0.654479\pi$
$269$	−15.3018	−0.932966	−0.466483	+	0.884530i	$0.654479\pi$
$270$	0.445154	0.0270912
$271$	31.7977	1.93157	0.965785	−	0.259343i	$-0.0835059\pi$
$271$	31.7977	1.93157	0.965785	+	0.259343i	$0.0835059\pi$
$272$	−8.90112	−0.539710
$273$	4.66486	0.282330
$274$	−0.838994	−0.0506855
$275$	−4.72592	−0.284984
$276$	12.1123	0.729073
$277$	2.90409	0.174490	0.0872450	−	0.996187i	$-0.472194\pi$
$277$	2.90409	0.174490	0.0872450	+	0.996187i	$0.472194\pi$
$278$	−0.163038	−0.00977835
$279$	5.18277	0.310284
$280$	3.76742	0.225147
$281$	−4.67668	−0.278987	−0.139494	−	0.990223i	$-0.544548\pi$
$281$	−4.67668	−0.278987	−0.139494	+	0.990223i	$0.544548\pi$
$282$	0.911064	0.0542531
$283$	−33.4138	−1.98624	−0.993122	−	0.117085i	$-0.962645\pi$
$283$	−33.4138	−1.98624	−0.993122	+	0.117085i	$0.962645\pi$
$284$	2.00739	0.119117
$285$	−11.0146	−0.652450
$286$	−0.528976	−0.0312790
$287$	18.0733	1.06683
$288$	−1.84995	−0.109009
$289$	−11.6637	−0.686102
$290$	0.595244	0.0349539
$291$	8.34884	0.489418
$292$	23.1664	1.35571
$293$	−0.0381493	−0.00222870	−0.00111435	−	0.999999i	$-0.500355\pi$
$293$	−0.0381493	−0.00222870	−0.00111435	+	0.999999i	$0.500355\pi$
$294$	−0.386773	−0.0225571
$295$	9.25851	0.539051
$296$	2.56597	0.149144
$297$	1.54035	0.0893804
$298$	−0.0605498	−0.00350755
$299$	−13.4354	−0.776991
$300$	6.06079	0.349920
$301$	−5.42287	−0.312569
$302$	−1.69427	−0.0974943
$303$	8.46921	0.486544
$304$	14.9421	0.856986
$305$	27.3234	1.56454
$306$	0.362030	0.0206959
$307$	−20.2291	−1.15453	−0.577267	−	0.816555i	$-0.695881\pi$
$307$	−20.2291	−1.15453	−0.577267	+	0.816555i	$0.695881\pi$
$308$	6.47788	0.369112
$309$	14.5237	0.826225
$310$	−2.30713	−0.131036
$311$	21.6980	1.23038	0.615191	−	0.788378i	$-0.289079\pi$
$311$	21.6980	1.23038	0.615191	+	0.788378i	$0.289079\pi$
$312$	1.36521	0.0772899
$313$	−30.2241	−1.70837	−0.854184	−	0.519971i	$-0.825943\pi$
$313$	−30.2241	−1.70837	−0.854184	+	0.519971i	$0.825943\pi$
$314$	1.46216	0.0825146
$315$	6.04692	0.340705
$316$	−5.36303	−0.301694
$317$	4.04047	0.226935	0.113468	−	0.993542i	$-0.463804\pi$
$317$	4.04047	0.226935	0.113468	+	0.993542i	$0.463804\pi$
$318$	2.21780	0.124368
$319$	2.05971	0.115321
$320$	−21.0662	−1.17764
$321$	−9.79683	−0.546806
$322$	−2.04568	−0.114001
$323$	−8.95786	−0.498428
$324$	−1.97544	−0.109747
$325$	−6.72288	−0.372918
$326$	2.19449	0.121541
$327$	10.4368	0.577157
$328$	5.28931	0.292053
$329$	12.3758	0.682298
$330$	−0.685695	−0.0377463
$331$	−32.3515	−1.77820	−0.889101	−	0.457711i	$-0.848669\pi$
$331$	−32.3515	−1.77820	−0.889101	+	0.457711i	$0.848669\pi$
$332$	−22.1848	−1.21755
$333$	4.11852	0.225694
$334$	−2.76225	−0.151144
$335$	−1.99050	−0.108753
$336$	−8.20303	−0.447512
$337$	23.5467	1.28267	0.641335	−	0.767261i	$-0.278381\pi$
$337$	23.5467	1.28267	0.641335	+	0.767261i	$0.278381\pi$
$338$	1.28487	0.0698877
$339$	12.0988	0.657114
$340$	12.9619	0.702956
$341$	−7.98330	−0.432320
$342$	−0.607730	−0.0328623
$343$	−20.1560	−1.08832
$344$	−1.58705	−0.0855680
$345$	−17.4159	−0.937643
$346$	−0.128572	−0.00691208
$347$	19.4827	1.04589	0.522944	−	0.852367i	$-0.324834\pi$
$347$	19.4827	1.04589	0.522944	+	0.852367i	$0.324834\pi$
$348$	−2.64149	−0.141598
$349$	24.5994	1.31678	0.658388	−	0.752679i	$-0.271239\pi$
$349$	24.5994	1.31678	0.658388	+	0.752679i	$0.271239\pi$
$350$	−1.02362	−0.0547150
$351$	2.19124	0.116960
$352$	2.84957	0.151883
$353$	−19.2017	−1.02200	−0.511000	−	0.859581i	$-0.670725\pi$
$353$	−19.2017	−1.02200	−0.511000	+	0.859581i	$0.670725\pi$
$354$	0.510836	0.0271506
$355$	−2.88638	−0.153193
$356$	19.4260	1.02958
$357$	4.91777	0.260276
$358$	0.560821	0.0296403
$359$	18.8736	0.996111	0.498056	−	0.867145i	$-0.334048\pi$
$359$	18.8736	0.996111	0.498056	+	0.867145i	$0.334048\pi$
$360$	1.76968	0.0932705
$361$	−3.96269	−0.208563
$362$	−0.768882	−0.0404115
$363$	8.62731	0.452816
$364$	9.21514	0.483005
$365$	−33.3104	−1.74355
$366$	1.50757	0.0788017
$367$	14.5027	0.757034	0.378517	−	0.925594i	$-0.376434\pi$
$367$	14.5027	0.757034	0.378517	+	0.925594i	$0.376434\pi$
$368$	23.6259	1.23158
$369$	8.48963	0.441952
$370$	−1.83338	−0.0953127
$371$	30.1264	1.56408
$372$	10.2382	0.530828
$373$	13.2613	0.686646	0.343323	−	0.939217i	$-0.388447\pi$
$373$	13.2613	0.686646	0.343323	+	0.939217i	$0.388447\pi$
$374$	−0.557654	−0.0288356
$375$	5.48751	0.283374
$376$	3.62188	0.186784
$377$	2.93004	0.150905
$378$	0.333637	0.0171604
$379$	14.3019	0.734639	0.367320	−	0.930095i	$-0.380276\pi$
$379$	14.3019	0.734639	0.367320	+	0.930095i	$0.380276\pi$
$380$	−21.7587	−1.11620
$381$	1.34751	0.0690350
$382$	−1.34553	−0.0688435
$383$	0.863316	0.0441134	0.0220567	−	0.999757i	$-0.492979\pi$
$383$	0.863316	0.0441134	0.0220567	+	0.999757i	$0.492979\pi$
$384$	−4.86222	−0.248124
$385$	−9.31439	−0.474705
$386$	−2.22893	−0.113450
$387$	−2.54730	−0.129486
$388$	16.4926	0.837286
$389$	33.7010	1.70871	0.854354	−	0.519691i	$-0.173953\pi$
$389$	33.7010	1.70871	0.854354	+	0.519691i	$0.173953\pi$
$390$	−0.975439	−0.0493933
$391$	−14.1638	−0.716296
$392$	−1.53759	−0.0776601
$393$	−11.5131	−0.580760
$394$	0.999477	0.0503529
$395$	7.71137	0.388001
$396$	3.04287	0.152910
$397$	−2.37071	−0.118983	−0.0594913	−	0.998229i	$-0.518948\pi$
$397$	−2.37071	−0.118983	−0.0594913	+	0.998229i	$0.518948\pi$
$398$	1.54792	0.0775904
$399$	−8.25532	−0.413283
$400$	11.8220	0.591100
$401$	17.0184	0.849858	0.424929	−	0.905227i	$-0.360299\pi$
$401$	17.0184	0.849858	0.424929	+	0.905227i	$0.360299\pi$
$402$	−0.109825	−0.00547759
$403$	−11.3567	−0.565717
$404$	16.7304	0.832369
$405$	2.84044	0.141142
$406$	0.446128	0.0221409
$407$	−6.34398	−0.314459
$408$	1.43923	0.0712524
$409$	−23.0341	−1.13896	−0.569481	−	0.822004i	$-0.692856\pi$
$409$	−23.0341	−1.13896	−0.569481	+	0.822004i	$0.692856\pi$
$410$	−3.77919	−0.186641
$411$	−5.35345	−0.264066
$412$	28.6907	1.41349
$413$	6.93913	0.341452
$414$	−0.960921	−0.0472267
$415$	31.8990	1.56586
$416$	4.05367	0.198748
$417$	−1.04031	−0.0509442
$418$	0.936119	0.0457871
$419$	−11.2079	−0.547542	−0.273771	−	0.961795i	$-0.588271\pi$
$419$	−11.2079	−0.547542	−0.273771	+	0.961795i	$0.588271\pi$
$420$	11.9453	0.582871
$421$	24.4933	1.19373	0.596865	−	0.802342i	$-0.296413\pi$
$421$	24.4933	1.19373	0.596865	+	0.802342i	$0.296413\pi$
$422$	−0.325194	−0.0158302
$423$	5.81331	0.282653
$424$	8.81675	0.428179
$425$	−7.08736	−0.343788
$426$	−0.159256	−0.00771596
$427$	20.4786	0.991027
$428$	−19.3530	−0.935464
$429$	−3.37528	−0.162960
$430$	1.13394	0.0546835
$431$	−4.20483	−0.202540	−0.101270	−	0.994859i	$-0.532291\pi$
$431$	−4.20483	−0.202540	−0.101270	+	0.994859i	$0.532291\pi$
$432$	−3.85324	−0.185389
$433$	−21.7297	−1.04426	−0.522131	−	0.852865i	$-0.674863\pi$
$433$	−21.7297	−1.04426	−0.522131	+	0.852865i	$0.674863\pi$
$434$	−1.72917	−0.0830026
$435$	3.79813	0.182106
$436$	20.6173	0.987389
$437$	23.7765	1.13738
$438$	−1.83790	−0.0878181
$439$	1.28769	0.0614579	0.0307290	−	0.999528i	$-0.490217\pi$
$439$	1.28769	0.0614579	0.0307290	+	0.999528i	$0.490217\pi$
$440$	−2.72594	−0.129954
$441$	−2.46792	−0.117520
$442$	−0.793294	−0.0377331
$443$	−16.8760	−0.801804	−0.400902	−	0.916121i	$-0.631303\pi$
$443$	−16.8760	−0.801804	−0.400902	+	0.916121i	$0.631303\pi$
$444$	8.13589	0.386112
$445$	−27.9322	−1.32411
$446$	−0.156720	−0.00742093
$447$	−0.386355	−0.0182740
$448$	−15.7889	−0.745953
$449$	−2.10530	−0.0993550	−0.0496775	−	0.998765i	$-0.515819\pi$
$449$	−2.10530	−0.0993550	−0.0496775	+	0.998765i	$0.515819\pi$
$450$	−0.480830	−0.0226665
$451$	−13.0770	−0.615773
$452$	23.9003	1.12418
$453$	−10.8108	−0.507935
$454$	0.0978635	0.00459296
$455$	−13.2502	−0.621180
$456$	−2.41599	−0.113139
$457$	−16.0391	−0.750279	−0.375140	−	0.926968i	$-0.622405\pi$
$457$	−16.0391	−0.750279	−0.375140	+	0.926968i	$0.622405\pi$
$458$	−3.56689	−0.166670
$459$	2.31004	0.107823
$460$	−34.4041	−1.60410
$461$	4.40905	0.205350	0.102675	−	0.994715i	$-0.467260\pi$
$461$	4.40905	0.205350	0.102675	+	0.994715i	$0.467260\pi$
$462$	−0.513920	−0.0239097
$463$	9.81635	0.456204	0.228102	−	0.973637i	$-0.426748\pi$
$463$	9.81635	0.456204	0.228102	+	0.973637i	$0.426748\pi$
$464$	−5.15241	−0.239195
$465$	−14.7213	−0.682685
$466$	2.14649	0.0994343
$467$	−36.2465	−1.67729	−0.838643	−	0.544681i	$-0.816651\pi$
$467$	−36.2465	−1.67729	−0.838643	+	0.544681i	$0.816651\pi$
$468$	4.32866	0.200092
$469$	−1.49185	−0.0688873
$470$	−2.58782	−0.119367
$471$	9.32975	0.429892
$472$	2.03080	0.0934750
$473$	3.92374	0.180414
$474$	0.425473	0.0195426
$475$	11.8974	0.545889
$476$	9.71475	0.445275
$477$	14.1513	0.647946
$478$	0.894012	0.0408911
$479$	−15.3898	−0.703178	−0.351589	−	0.936154i	$-0.614359\pi$
$479$	−15.3898	−0.703178	−0.351589	+	0.936154i	$0.614359\pi$
$480$	5.25465	0.239841
$481$	−9.02466	−0.411489
$482$	4.08548	0.186089
$483$	−13.0530	−0.593933
$484$	17.0427	0.774669
$485$	−23.7143	−1.07681
$486$	0.156720	0.00710898
$487$	−23.4118	−1.06089	−0.530445	−	0.847719i	$-0.677975\pi$
$487$	−23.4118	−1.06089	−0.530445	+	0.847719i	$0.677975\pi$
$488$	5.99323	0.271301
$489$	14.0026	0.633218
$490$	1.09860	0.0496298
$491$	27.3735	1.23535	0.617675	−	0.786434i	$-0.288075\pi$
$491$	27.3735	1.23535	0.617675	+	0.786434i	$0.288075\pi$
$492$	16.7707	0.756083
$493$	3.08890	0.139117
$494$	1.33168	0.0599151
$495$	−4.37528	−0.196654
$496$	19.9704	0.896699
$497$	−2.16331	−0.0970375
$498$	1.76002	0.0788684
$499$	−7.38480	−0.330589	−0.165295	−	0.986244i	$-0.552857\pi$
$499$	−7.38480	−0.330589	−0.165295	+	0.986244i	$0.552857\pi$
$500$	10.8402	0.484791
$501$	−17.6253	−0.787442
$502$	2.04104	0.0910960
$503$	10.7944	0.481299	0.240650	−	0.970612i	$-0.422639\pi$
$503$	10.7944	0.481299	0.240650	+	0.970612i	$0.422639\pi$
$504$	1.32635	0.0590805
$505$	−24.0563	−1.07049
$506$	1.48016	0.0658011
$507$	8.19848	0.364107
$508$	2.66192	0.118104
$509$	−6.92728	−0.307046	−0.153523	−	0.988145i	$-0.549062\pi$
$509$	−6.92728	−0.307046	−0.153523	+	0.988145i	$0.549062\pi$
$510$	−1.02832	−0.0455349
$511$	−24.9657	−1.10442
$512$	−11.9297	−0.527221
$513$	−3.87780	−0.171209
$514$	1.80616	0.0796663
$515$	−41.2537	−1.81785
$516$	−5.03203	−0.221523
$517$	−8.95455	−0.393821
$518$	−1.37409	−0.0603741
$519$	−0.820392	−0.0360112
$520$	−3.87780	−0.170053
$521$	24.2501	1.06241	0.531207	−	0.847242i	$-0.321739\pi$
$521$	24.2501	1.06241	0.531207	+	0.847242i	$0.321739\pi$
$522$	0.209561	0.00917223
$523$	−10.4895	−0.458676	−0.229338	−	0.973347i	$-0.573656\pi$
$523$	−10.4895	−0.458676	−0.229338	+	0.973347i	$0.573656\pi$
$524$	−22.7435	−0.993552
$525$	−6.53153	−0.285059
$526$	2.77702	0.121084
$527$	−11.9724	−0.521526
$528$	5.93535	0.258303
$529$	14.5945	0.634543
$530$	−6.29953	−0.273634
$531$	3.25954	0.141452
$532$	−16.3079	−0.707036
$533$	−18.6028	−0.805776
$534$	−1.54115	−0.0666921
$535$	27.8273	1.20308
$536$	−0.436604	−0.0188584
$537$	3.57848	0.154423
$538$	2.39810	0.103390
$539$	3.80146	0.163741
$540$	5.61111	0.241464
$541$	−27.8694	−1.19820	−0.599099	−	0.800675i	$-0.704474\pi$
$541$	−27.8694	−1.19820	−0.599099	+	0.800675i	$0.704474\pi$
$542$	−4.98334	−0.214053
$543$	−4.90607	−0.210540
$544$	4.27344	0.183222
$545$	−29.6451	−1.26986
$546$	−0.731079	−0.0312873
$547$	−17.6178	−0.753284	−0.376642	−	0.926359i	$-0.622921\pi$
$547$	−17.6178	−0.753284	−0.376642	+	0.926359i	$0.622921\pi$
$548$	−10.5754	−0.451759
$549$	9.61946	0.410548
$550$	0.740648	0.0315813
$551$	−5.18525	−0.220899
$552$	−3.82008	−0.162594
$553$	5.77957	0.245772
$554$	−0.455131	−0.0193366
$555$	−11.6984	−0.496569
$556$	−2.05507	−0.0871543
$557$	−26.7268	−1.13245	−0.566225	−	0.824251i	$-0.691597\pi$
$557$	−26.7268	−1.13245	−0.566225	+	0.824251i	$0.691597\pi$
$558$	−0.812246	−0.0343851
$559$	5.58174	0.236082
$560$	23.3002	0.984613
$561$	−3.55828	−0.150230
$562$	0.732932	0.0309169
$563$	−12.6240	−0.532038	−0.266019	−	0.963968i	$-0.585708\pi$
$563$	−12.6240	−0.532038	−0.266019	+	0.963968i	$0.585708\pi$
$564$	11.4838	0.483557
$565$	−34.3657	−1.44578
$566$	5.23662	0.220112
$567$	2.12887	0.0894041
$568$	−0.633111	−0.0265647
$569$	33.1890	1.39136	0.695678	−	0.718354i	$-0.255104\pi$
$569$	33.1890	1.39136	0.695678	+	0.718354i	$0.255104\pi$
$570$	1.72622	0.0723033
$571$	29.3475	1.22816	0.614078	−	0.789246i	$-0.289528\pi$
$571$	29.3475	1.22816	0.614078	+	0.789246i	$0.289528\pi$
$572$	−6.66766	−0.278789
$573$	−8.58557	−0.358667
$574$	−2.83246	−0.118224
$575$	18.8117	0.784502
$576$	−7.41655	−0.309023
$577$	−17.7792	−0.740157	−0.370079	−	0.929000i	$-0.620669\pi$
$577$	−17.7792	−0.740157	−0.370079	+	0.929000i	$0.620669\pi$
$578$	1.82794	0.0760325
$579$	−14.2223	−0.591060
$580$	7.50297	0.311544
$581$	23.9079	0.991867
$582$	−1.30843	−0.0542363
$583$	−21.7981	−0.902784
$584$	−7.30644	−0.302343
$585$	−6.22407	−0.257334
$586$	0.00597877	0.000246981 0
$587$	16.0416	0.662106	0.331053	−	0.943612i	$-0.392596\pi$
$587$	16.0416	0.662106	0.331053	+	0.943612i	$0.392596\pi$
$588$	−4.87522	−0.201051
$589$	20.0977	0.828113
$590$	−1.45100	−0.0597366
$591$	6.37745	0.262333
$592$	15.8696	0.652238
$593$	−7.41248	−0.304394	−0.152197	−	0.988350i	$-0.548635\pi$
$593$	−7.41248	−0.304394	−0.152197	+	0.988350i	$0.548635\pi$
$594$	−0.241405	−0.00990496
$595$	−13.9686	−0.572657
$596$	−0.763221	−0.0312628
$597$	9.87698	0.404238
$598$	2.10561	0.0861046
$599$	3.17987	0.129926	0.0649630	−	0.997888i	$-0.479307\pi$
$599$	3.17987	0.129926	0.0649630	+	0.997888i	$0.479307\pi$
$600$	−1.91151	−0.0780370
$601$	10.0473	0.409837	0.204918	−	0.978779i	$-0.434307\pi$
$601$	10.0473	0.409837	0.204918	+	0.978779i	$0.434307\pi$
$602$	0.849874	0.0346383
$603$	−0.700772	−0.0285376
$604$	−21.3560	−0.868965
$605$	−24.5053	−0.996283
$606$	−1.32730	−0.0539178
$607$	29.3393	1.19084	0.595422	−	0.803413i	$-0.296985\pi$
$607$	29.3393	1.19084	0.595422	+	0.803413i	$0.296985\pi$
$608$	−7.17371	−0.290932
$609$	2.84665	0.115352
$610$	−4.28214	−0.173379
$611$	−12.7383	−0.515338
$612$	4.56334	0.184462
$613$	13.6002	0.549308	0.274654	−	0.961543i	$-0.411437\pi$
$613$	13.6002	0.549308	0.274654	+	0.961543i	$0.411437\pi$
$614$	3.17031	0.127943
$615$	−24.1142	−0.972380
$616$	−2.04306	−0.0823171
$617$	−42.3473	−1.70484	−0.852420	−	0.522858i	$-0.824866\pi$
$617$	−42.3473	−1.70484	−0.852420	+	0.522858i	$0.824866\pi$
$618$	−2.27616	−0.0915607
$619$	48.0817	1.93257	0.966284	−	0.257478i	$-0.0828914\pi$
$619$	48.0817	1.93257	0.966284	+	0.257478i	$0.0828914\pi$
$620$	−29.0811	−1.16792
$621$	−6.13143	−0.246046
$622$	−3.40052	−0.136348
$623$	−20.9348	−0.838735
$624$	8.44335	0.338005
$625$	−30.9273	−1.23709
$626$	4.73674	0.189318
$627$	5.97318	0.238546
$628$	18.4303	0.735451
$629$	−9.51394	−0.379346
$630$	−0.947675	−0.0377563
$631$	−44.6639	−1.77804	−0.889020	−	0.457868i	$-0.848613\pi$
$631$	−44.6639	−1.77804	−0.889020	+	0.457868i	$0.848613\pi$
$632$	1.69144	0.0672820
$633$	−2.07499	−0.0824736
$634$	−0.633224	−0.0251485
$635$	−3.82751	−0.151890
$636$	27.9551	1.10849
$637$	5.40779	0.214264
$638$	−0.322798	−0.0127797
$639$	−1.01618	−0.0401993
$640$	13.8108	0.545920
$641$	48.6311	1.92081	0.960407	−	0.278600i	$-0.0898703\pi$
$641$	48.6311	1.92081	0.960407	+	0.278600i	$0.0898703\pi$
$642$	1.53536	0.0605960
$643$	33.1794	1.30847	0.654233	−	0.756293i	$-0.272992\pi$
$643$	33.1794	1.30847	0.654233	+	0.756293i	$0.272992\pi$
$644$	−25.7854	−1.01609
$645$	7.23544	0.284895
$646$	1.40388	0.0552349
$647$	−46.7208	−1.83679	−0.918393	−	0.395669i	$-0.870513\pi$
$647$	−46.7208	−1.83679	−0.918393	+	0.395669i	$0.870513\pi$
$648$	0.623032	0.0244750
$649$	−5.02084	−0.197085
$650$	1.05361	0.0413261
$651$	−11.0334	−0.432435
$652$	27.6612	1.08330
$653$	16.1343	0.631386	0.315693	−	0.948861i	$-0.397763\pi$
$653$	16.1343	0.631386	0.315693	+	0.948861i	$0.397763\pi$
$654$	−1.63566	−0.0639594
$655$	32.7023	1.27778
$656$	32.7125	1.27721
$657$	−11.7272	−0.457523
$658$	−1.93954	−0.0756110
$659$	26.1096	1.01709	0.508543	−	0.861037i	$-0.330184\pi$
$659$	26.1096	1.01709	0.508543	+	0.861037i	$0.330184\pi$
$660$	−8.64309	−0.336432
$661$	43.0870	1.67589	0.837944	−	0.545756i	$-0.183757\pi$
$661$	43.0870	1.67589	0.837944	+	0.545756i	$0.183757\pi$
$662$	5.07015	0.197057
$663$	−5.06184	−0.196586
$664$	6.99686	0.271531
$665$	23.4487	0.909302
$666$	−0.645456	−0.0250109
$667$	−8.19873	−0.317456
$668$	−34.8178	−1.34714
$669$	−1.00000	−0.0386622
$670$	0.311952	0.0120517
$671$	−14.8174	−0.572018
$672$	3.93829	0.151923
$673$	−24.2241	−0.933770	−0.466885	−	0.884318i	$-0.654624\pi$
$673$	−24.2241	−0.933770	−0.466885	+	0.884318i	$0.654624\pi$
$674$	−3.69025	−0.142143
$675$	−3.06807	−0.118090
$676$	16.1956	0.622907
$677$	6.29496	0.241935	0.120968	−	0.992656i	$-0.461400\pi$
$677$	6.29496	0.241935	0.120968	+	0.992656i	$0.461400\pi$
$678$	−1.89612	−0.0728201
$679$	−17.7736	−0.682088
$680$	−4.08803	−0.156769
$681$	0.624446	0.0239288
$682$	1.25115	0.0479089
$683$	−31.3528	−1.19968	−0.599840	−	0.800120i	$-0.704769\pi$
$683$	−31.3528	−1.19968	−0.599840	+	0.800120i	$0.704769\pi$
$684$	−7.66035	−0.292901
$685$	15.2061	0.580996
$686$	3.15885	0.120605
$687$	−22.7596	−0.868331
$688$	−9.81534	−0.374206
$689$	−31.0090	−1.18135
$690$	2.72943	0.103908
$691$	4.09065	0.155616	0.0778079	−	0.996968i	$-0.475208\pi$
$691$	4.09065	0.155616	0.0778079	+	0.996968i	$0.475208\pi$
$692$	−1.62063	−0.0616073
$693$	−3.27921	−0.124567
$694$	−3.05334	−0.115903
$695$	2.95493	0.112087
$696$	0.833097	0.0315784
$697$	−19.6114	−0.742833
$698$	−3.85523	−0.145922
$699$	13.6963	0.518042
$700$	−12.9026	−0.487674
$701$	−3.29031	−0.124273	−0.0621367	−	0.998068i	$-0.519791\pi$
$701$	−3.29031	−0.124273	−0.0621367	+	0.998068i	$0.519791\pi$
$702$	−0.343412	−0.0129612
$703$	15.9708	0.602350
$704$	11.4241	0.430562
$705$	−16.5123	−0.621890
$706$	3.00929	0.113256
$707$	−18.0298	−0.678082
$708$	6.43902	0.241993
$709$	−44.2140	−1.66049	−0.830246	−	0.557397i	$-0.811800\pi$
$709$	−44.2140	−1.66049	−0.830246	+	0.557397i	$0.811800\pi$
$710$	0.452355	0.0169766
$711$	2.71486	0.101815
$712$	−6.12675	−0.229610
$713$	31.7778	1.19009
$714$	−0.770715	−0.0288433
$715$	9.58727	0.358544
$716$	7.06907	0.264184
$717$	5.70450	0.213038
$718$	−2.95788	−0.110387
$719$	2.88332	0.107530	0.0537648	−	0.998554i	$-0.482878\pi$
$719$	2.88332	0.107530	0.0537648	+	0.998554i	$0.482878\pi$
$720$	10.9449	0.407891
$721$	−30.9191	−1.15149
$722$	0.621035	0.0231125
$723$	26.0686	0.969502
$724$	−9.69165	−0.360187
$725$	−4.10252	−0.152364
$726$	−1.35208	−0.0501802
$727$	−2.75667	−0.102239	−0.0511196	−	0.998693i	$-0.516279\pi$
$727$	−2.75667	−0.102239	−0.0511196	+	0.998693i	$0.516279\pi$
$728$	−2.90636	−0.107717
$729$	1.00000	0.0370370
$730$	5.22042	0.193217
$731$	5.88436	0.217641
$732$	19.0026	0.702358
$733$	51.1848	1.89055	0.945276	−	0.326272i	$-0.105792\pi$
$733$	51.1848	1.89055	0.945276	+	0.326272i	$0.105792\pi$
$734$	−2.27287	−0.0838930
$735$	7.00996	0.258566
$736$	−11.3428	−0.418102
$737$	1.07944	0.0397616
$738$	−1.33050	−0.0489763
$739$	−11.2989	−0.415636	−0.207818	−	0.978168i	$-0.566636\pi$
$739$	−11.2989	−0.415636	−0.207818	+	0.978168i	$0.566636\pi$
$740$	−23.1095	−0.849521
$741$	8.49717	0.312151
$742$	−4.72142	−0.173329
$743$	−41.9624	−1.53945	−0.769725	−	0.638376i	$-0.779607\pi$
$743$	−41.9624	−1.53945	−0.769725	+	0.638376i	$0.779607\pi$
$744$	−3.22904	−0.118382
$745$	1.09742	0.0402063
$746$	−2.07832	−0.0760928
$747$	11.2303	0.410896
$748$	−7.02915	−0.257011
$749$	20.8562	0.762068
$750$	−0.860005	−0.0314030
$751$	4.22921	0.154326	0.0771630	−	0.997018i	$-0.475414\pi$
$751$	4.22921	0.154326	0.0771630	+	0.997018i	$0.475414\pi$
$752$	22.4000	0.816846
$753$	13.0234	0.474601
$754$	−0.459198	−0.0167230
$755$	30.7073	1.11755
$756$	4.20545	0.152951
$757$	35.9230	1.30564	0.652821	−	0.757512i	$-0.273585\pi$
$757$	35.9230	1.30564	0.652821	+	0.757512i	$0.273585\pi$
$758$	−2.24140	−0.0814113
$759$	9.44458	0.342817
$760$	6.86247	0.248928
$761$	26.2321	0.950911	0.475456	−	0.879740i	$-0.342283\pi$
$761$	26.2321	0.950911	0.475456	+	0.879740i	$0.342283\pi$
$762$	−0.211182	−0.00765033
$763$	−22.2186	−0.804368
$764$	−16.9603	−0.613601
$765$	−6.56151	−0.237232
$766$	−0.135299	−0.00488856
$767$	−7.14242	−0.257898
$768$	−14.0711	−0.507747
$769$	−16.0013	−0.577022	−0.288511	−	0.957477i	$-0.593160\pi$
$769$	−16.0013	−0.577022	−0.288511	+	0.957477i	$0.593160\pi$
$770$	1.45976	0.0526059
$771$	11.5247	0.415053
$772$	−28.0954	−1.01117
$773$	−15.8343	−0.569522	−0.284761	−	0.958599i	$-0.591914\pi$
$773$	−15.8343	−0.569522	−0.284761	+	0.958599i	$0.591914\pi$
$774$	0.399214	0.0143494
$775$	15.9011	0.571185
$776$	−5.20160	−0.186726
$777$	−8.76779	−0.314543
$778$	−5.28164	−0.189356
$779$	32.9210	1.17952
$780$	−12.2953	−0.440241
$781$	1.56527	0.0560098
$782$	2.21976	0.0793786
$783$	1.33716	0.0477863
$784$	−9.50946	−0.339624
$785$	−26.5005	−0.945845
$786$	1.80434	0.0643587
$787$	−21.0443	−0.750149	−0.375074	−	0.926995i	$-0.622383\pi$
$787$	−21.0443	−0.750149	−0.375074	+	0.926995i	$0.622383\pi$
$788$	12.5983	0.448795
$789$	17.7196	0.630834
$790$	−1.20853	−0.0429976
$791$	−25.7567	−0.915801
$792$	−0.959690	−0.0341011
$793$	−21.0785	−0.748520
$794$	0.371539	0.0131854
$795$	−40.1960	−1.42560
$796$	19.5114	0.691562
$797$	34.8666	1.23504	0.617520	−	0.786555i	$-0.288137\pi$
$797$	34.8666	1.23504	0.617520	+	0.786555i	$0.288137\pi$
$798$	1.29378	0.0457992
$799$	−13.4290	−0.475082
$800$	−5.67577	−0.200669
$801$	−9.83376	−0.347459
$802$	−2.66713	−0.0941797
$803$	18.0641	0.637468
$804$	−1.38433	−0.0488216
$805$	37.0763	1.30677
$806$	1.77982	0.0626916
$807$	15.3018	0.538648
$808$	−5.27659	−0.185630
$809$	−3.49115	−0.122742	−0.0613711	−	0.998115i	$-0.519547\pi$
$809$	−3.49115	−0.122742	−0.0613711	+	0.998115i	$0.519547\pi$
$810$	−0.445154	−0.0156411
$811$	32.5362	1.14250	0.571251	−	0.820775i	$-0.306458\pi$
$811$	32.5362	1.14250	0.571251	+	0.820775i	$0.306458\pi$
$812$	5.62338	0.197342
$813$	−31.7977	−1.11519
$814$	0.994231	0.0348478
$815$	−39.7734	−1.39320
$816$	8.90112	0.311601
$817$	−9.87791	−0.345584
$818$	3.60991	0.126218
$819$	−4.66486	−0.163003
$820$	−47.6362	−1.66353
$821$	16.8367	0.587603	0.293802	−	0.955866i	$-0.405079\pi$
$821$	16.8367	0.587603	0.293802	+	0.955866i	$0.405079\pi$
$822$	0.838994	0.0292633
$823$	−50.9253	−1.77515	−0.887573	−	0.460667i	$-0.847610\pi$
$823$	−50.9253	−1.77515	−0.887573	+	0.460667i	$0.847610\pi$
$824$	−9.04875	−0.315228
$825$	4.72592	0.164535
$826$	−1.08750	−0.0378391
$827$	−40.8458	−1.42035	−0.710173	−	0.704027i	$-0.751383\pi$
$827$	−40.8458	−1.42035	−0.710173	+	0.704027i	$0.751383\pi$
$828$	−12.1123	−0.420931
$829$	15.8246	0.549610	0.274805	−	0.961500i	$-0.411387\pi$
$829$	15.8246	0.549610	0.274805	+	0.961500i	$0.411387\pi$
$830$	−4.99923	−0.173526
$831$	−2.90409	−0.100742
$832$	16.2514	0.563416
$833$	5.70098	0.197527
$834$	0.163038	0.00564554
$835$	50.0637	1.73252
$836$	11.7997	0.408099
$837$	−5.18277	−0.179143
$838$	1.75651	0.0606776
$839$	−8.20745	−0.283353	−0.141676	−	0.989913i	$-0.545249\pi$
$839$	−8.20745	−0.283353	−0.141676	+	0.989913i	$0.545249\pi$
$840$	−3.76742	−0.129988
$841$	−27.2120	−0.938345
$842$	−3.83860	−0.132287
$843$	4.67668	0.161073
$844$	−4.09902	−0.141094
$845$	−23.2872	−0.801106
$846$	−0.911064	−0.0313230
$847$	−18.3664	−0.631077
$848$	54.5285	1.87252
$849$	33.4138	1.14676
$850$	1.11073	0.0380979
$851$	25.2524	0.865643
$852$	−2.00739	−0.0687722
$853$	−40.3568	−1.38179	−0.690895	−	0.722955i	$-0.742783\pi$
$853$	−40.3568	−1.38179	−0.690895	+	0.722955i	$0.742783\pi$
$854$	−3.20941	−0.109824
$855$	11.0146	0.376692
$856$	6.10374	0.208622
$857$	46.2414	1.57958	0.789789	−	0.613379i	$-0.210190\pi$
$857$	46.2414	1.57958	0.789789	+	0.613379i	$0.210190\pi$
$858$	0.528976	0.0180589
$859$	30.2294	1.03141	0.515707	−	0.856765i	$-0.327529\pi$
$859$	30.2294	1.03141	0.515707	+	0.856765i	$0.327529\pi$
$860$	14.2932	0.487393
$861$	−18.0733	−0.615937
$862$	0.658983	0.0224451
$863$	28.9509	0.985500	0.492750	−	0.870171i	$-0.335992\pi$
$863$	28.9509	0.985500	0.492750	+	0.870171i	$0.335992\pi$
$864$	1.84995	0.0629364
$865$	2.33027	0.0792316
$866$	3.40548	0.115723
$867$	11.6637	0.396121
$868$	−21.7959	−0.739801
$869$	−4.18184	−0.141859
$870$	−0.595244	−0.0201807
$871$	1.53556	0.0520304
$872$	−6.50247	−0.220201
$873$	−8.34884	−0.282565
$874$	−3.72626	−0.126043
$875$	−11.6822	−0.394930
$876$	−23.1664	−0.782721
$877$	−32.0836	−1.08339	−0.541694	−	0.840576i	$-0.682217\pi$
$877$	−32.0836	−1.08339	−0.541694	+	0.840576i	$0.682217\pi$
$878$	−0.201807	−0.00681065
$879$	0.0381493	0.00128674
$880$	−16.8590	−0.568316
$881$	−29.3594	−0.989142	−0.494571	−	0.869137i	$-0.664675\pi$
$881$	−29.3594	−0.989142	−0.494571	+	0.869137i	$0.664675\pi$
$882$	0.386773	0.0130233
$883$	−3.21324	−0.108134	−0.0540671	−	0.998537i	$-0.517219\pi$
$883$	−3.21324	−0.108134	−0.0540671	+	0.998537i	$0.517219\pi$
$884$	−9.99936	−0.336315
$885$	−9.25851	−0.311221
$886$	2.64482	0.0888544
$887$	45.7389	1.53576	0.767881	−	0.640593i	$-0.221311\pi$
$887$	45.7389	1.53576	0.767881	+	0.640593i	$0.221311\pi$
$888$	−2.56597	−0.0861084
$889$	−2.86867	−0.0962122
$890$	4.37754	0.146736
$891$	−1.54035	−0.0516038
$892$	−1.97544	−0.0661426
$893$	22.5428	0.754367
$894$	0.0605498	0.00202509
$895$	−10.1644	−0.339760
$896$	10.3510	0.345803
$897$	13.4354	0.448596
$898$	0.329943	0.0110103
$899$	−6.93022	−0.231136
$900$	−6.06079	−0.202026
$901$	−32.6901	−1.08907
$902$	2.04944	0.0682388
$903$	5.42287	0.180462
$904$	−7.53792	−0.250707
$905$	13.9354	0.463228
$906$	1.69427	0.0562884
$907$	−15.0541	−0.499863	−0.249932	−	0.968263i	$-0.580408\pi$
$907$	−15.0541	−0.499863	−0.249932	+	0.968263i	$0.580408\pi$
$908$	1.23356	0.0409370
$909$	−8.46921	−0.280906
$910$	2.07658	0.0688380
$911$	−16.7890	−0.556243	−0.278121	−	0.960546i	$-0.589712\pi$
$911$	−16.7890	−0.556243	−0.278121	+	0.960546i	$0.589712\pi$
$912$	−14.9421	−0.494781
$913$	−17.2987	−0.572503
$914$	2.51366	0.0831445
$915$	−27.3234	−0.903285
$916$	−44.9601	−1.48552
$917$	24.5099	0.809389
$918$	−0.362030	−0.0119488
$919$	23.5577	0.777095	0.388548	−	0.921429i	$-0.372977\pi$
$919$	23.5577	0.777095	0.388548	+	0.921429i	$0.372977\pi$
$920$	10.8507	0.357737
$921$	20.2291	0.666571
$922$	−0.690989	−0.0227565
$923$	2.22668	0.0732922
$924$	−6.47788	−0.213107
$925$	12.6359	0.415467
$926$	−1.53842	−0.0505557
$927$	−14.5237	−0.477021
$928$	2.47368	0.0812025
$929$	55.2016	1.81111	0.905553	−	0.424234i	$-0.139457\pi$
$929$	55.2016	1.81111	0.905553	+	0.424234i	$0.139457\pi$
$930$	2.30713	0.0756539
$931$	−9.57008	−0.313647
$932$	27.0562	0.886256
$933$	−21.6980	−0.710361
$934$	5.68056	0.185874
$935$	10.1071	0.330536
$936$	−1.36521	−0.0446234
$937$	54.4519	1.77887	0.889433	−	0.457066i	$-0.151100\pi$
$937$	54.4519	1.77887	0.889433	+	0.457066i	$0.151100\pi$
$938$	0.233804	0.00763396
$939$	30.2241	0.986327
$940$	−32.6191	−1.06392
$941$	2.78328	0.0907324	0.0453662	−	0.998970i	$-0.485555\pi$
$941$	2.78328	0.0907324	0.0453662	+	0.998970i	$0.485555\pi$
$942$	−1.46216	−0.0476398
$943$	52.0536	1.69510
$944$	12.5598	0.408786
$945$	−6.04692	−0.196706
$946$	−0.614930	−0.0199931
$947$	−12.5947	−0.409273	−0.204636	−	0.978838i	$-0.565601\pi$
$947$	−12.5947	−0.409273	−0.204636	+	0.978838i	$0.565601\pi$
$948$	5.36303	0.174183
$949$	25.6971	0.834164
$950$	−1.86456	−0.0604943
$951$	−4.04047	−0.131021
$952$	−3.06393	−0.0993025
$953$	37.1771	1.20428	0.602142	−	0.798389i	$-0.294314\pi$
$953$	37.1771	1.20428	0.602142	+	0.798389i	$0.294314\pi$
$954$	−2.21780	−0.0718041
$955$	24.3867	0.789137
$956$	11.2689	0.364462
$957$	−2.05971	−0.0665808
$958$	2.41190	0.0779248
$959$	11.3968	0.368021
$960$	21.0662	0.679909
$961$	−4.13887	−0.133512
$962$	1.41435	0.0456004
$963$	9.79683	0.315698
$964$	51.4969	1.65860
$965$	40.3976	1.30045
$966$	2.04568	0.0658185
$967$	26.5973	0.855312	0.427656	−	0.903941i	$-0.359339\pi$
$967$	26.5973	0.855312	0.427656	+	0.903941i	$0.359339\pi$
$968$	−5.37509	−0.172762
$969$	8.95786	0.287768
$970$	3.71652	0.119330
$971$	23.1102	0.741641	0.370821	−	0.928704i	$-0.379076\pi$
$971$	23.1102	0.741641	0.370821	+	0.928704i	$0.379076\pi$
$972$	1.97544	0.0633622
$973$	2.21468	0.0709995
$974$	3.66911	0.117566
$975$	6.72288	0.215304
$976$	37.0660	1.18645
$977$	30.7443	0.983599	0.491799	−	0.870709i	$-0.336339\pi$
$977$	30.7443	0.983599	0.491799	+	0.870709i	$0.336339\pi$
$978$	−2.19449	−0.0701720
$979$	15.1475	0.484115
$980$	13.8477	0.442350
$981$	−10.4368	−0.333222
$982$	−4.28999	−0.136899
$983$	26.6052	0.848575	0.424287	−	0.905528i	$-0.360525\pi$
$983$	26.6052	0.848575	0.424287	+	0.905528i	$0.360525\pi$
$984$	−5.28931	−0.168617
$985$	−18.1147	−0.577184
$986$	−0.484094	−0.0154167
$987$	−12.3758	−0.393925
$988$	16.7856	0.534023
$989$	−15.6186	−0.496642
$990$	0.685695	0.0217928
$991$	−20.9542	−0.665632	−0.332816	−	0.942992i	$-0.607999\pi$
$991$	−20.9542	−0.665632	−0.332816	+	0.942992i	$0.607999\pi$
$992$	−9.58785	−0.304414
$993$	32.3515	1.02665
$994$	0.339034	0.0107535
$995$	−28.0549	−0.889401
$996$	22.1848	0.702953
$997$	−1.04005	−0.0329386	−0.0164693	−	0.999864i	$-0.505243\pi$
$997$	−1.04005	−0.0329386	−0.0164693	+	0.999864i	$0.505243\pi$
$998$	1.15735	0.0366352
$999$	−4.11852	−0.130304

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	669.2.a.h.1.3	✓	7
3.2	odd	2		2007.2.a.l.1.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
669.2.a.h.1.3	✓	7	1.1	even	1	trivial
2007.2.a.l.1.5		7	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 669 = 3 \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	669.a (trivial)

\(f(q)\)	\(=\)	\(q-0.156720 q^{2} -1.00000 q^{3} -1.97544 q^{4} +2.84044 q^{5} +0.156720 q^{6} +2.12887 q^{7} +0.623032 q^{8} +1.00000 q^{9} -0.445154 q^{10} -1.54035 q^{11} +1.97544 q^{12} -2.19124 q^{13} -0.333637 q^{14} -2.84044 q^{15} +3.85324 q^{16} -2.31004 q^{17} -0.156720 q^{18} +3.87780 q^{19} -5.61111 q^{20} -2.12887 q^{21} +0.241405 q^{22} +6.13143 q^{23} -0.623032 q^{24} +3.06807 q^{25} +0.343412 q^{26} -1.00000 q^{27} -4.20545 q^{28} -1.33716 q^{29} +0.445154 q^{30} +5.18277 q^{31} -1.84995 q^{32} +1.54035 q^{33} +0.362030 q^{34} +6.04692 q^{35} -1.97544 q^{36} +4.11852 q^{37} -0.607730 q^{38} +2.19124 q^{39} +1.76968 q^{40} +8.48963 q^{41} +0.333637 q^{42} -2.54730 q^{43} +3.04287 q^{44} +2.84044 q^{45} -0.960921 q^{46} +5.81331 q^{47} -3.85324 q^{48} -2.46792 q^{49} -0.480830 q^{50} +2.31004 q^{51} +4.32866 q^{52} +14.1513 q^{53} +0.156720 q^{54} -4.37528 q^{55} +1.32635 q^{56} -3.87780 q^{57} +0.209561 q^{58} +3.25954 q^{59} +5.61111 q^{60} +9.61946 q^{61} -0.812246 q^{62} +2.12887 q^{63} -7.41655 q^{64} -6.22407 q^{65} -0.241405 q^{66} -0.700772 q^{67} +4.56334 q^{68} -6.13143 q^{69} -0.947675 q^{70} -1.01618 q^{71} +0.623032 q^{72} -11.7272 q^{73} -0.645456 q^{74} -3.06807 q^{75} -7.66035 q^{76} -3.27921 q^{77} -0.343412 q^{78} +2.71486 q^{79} +10.9449 q^{80} +1.00000 q^{81} -1.33050 q^{82} +11.2303 q^{83} +4.20545 q^{84} -6.56151 q^{85} +0.399214 q^{86} +1.33716 q^{87} -0.959690 q^{88} -9.83376 q^{89} -0.445154 q^{90} -4.66486 q^{91} -12.1123 q^{92} -5.18277 q^{93} -0.911064 q^{94} +11.0146 q^{95} +1.84995 q^{96} -8.34884 q^{97} +0.386773 q^{98} -1.54035 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{2} - 7 q^{3} + 6 q^{4} + 3 q^{5} - 2 q^{6} - 2 q^{7} + 3 q^{8} + 7 q^{9} + 15 q^{11} - 6 q^{12} - 2 q^{13} + 4 q^{14} - 3 q^{15} + 5 q^{17} + 2 q^{18} + 20 q^{19} + 8 q^{20} + 2 q^{21} + 11 q^{22}+ \cdots + 15 q^{99}+O(q^{100}) \) 7 * q + 2 * q^2 - 7 * q^3 + 6 * q^4 + 3 * q^5 - 2 * q^6 - 2 * q^7 + 3 * q^8 + 7 * q^9 + 15 * q^11 - 6 * q^12 - 2 * q^13 + 4 * q^14 - 3 * q^15 + 5 * q^17 + 2 * q^18 + 20 * q^19 + 8 * q^20 + 2 * q^21 + 11 * q^22 + 9 * q^23 - 3 * q^24 - 4 * q^25 + 19 * q^26 - 7 * q^27 - 8 * q^28 + 9 * q^29 + 10 * q^31 + 8 * q^32 - 15 * q^33 + q^34 + 12 * q^35 + 6 * q^36 - 16 * q^37 + 25 * q^38 + 2 * q^39 + 17 * q^40 + 16 * q^41 - 4 * q^42 - 5 * q^43 + 26 * q^44 + 3 * q^45 + 2 * q^46 + 21 * q^47 + 3 * q^49 + 6 * q^50 - 5 * q^51 - 9 * q^52 + 5 * q^53 - 2 * q^54 + 7 * q^55 + 14 * q^56 - 20 * q^57 + 2 * q^58 + 26 * q^59 - 8 * q^60 + 12 * q^61 + q^62 - 2 * q^63 - 41 * q^64 - 4 * q^65 - 11 * q^66 - 3 * q^67 + 8 * q^68 - 9 * q^69 - 34 * q^70 + 21 * q^71 + 3 * q^72 - 23 * q^73 - 10 * q^74 + 4 * q^75 + 35 * q^76 - 16 * q^77 - 19 * q^78 + 13 * q^79 - 4 * q^80 + 7 * q^81 - 15 * q^82 + 26 * q^83 + 8 * q^84 - 25 * q^85 - 44 * q^86 - 9 * q^87 + 15 * q^88 + 20 * q^89 + 30 * q^91 - 5 * q^92 - 10 * q^93 + 5 * q^94 + 10 * q^95 - 8 * q^96 - 9 * q^97 - 32 * q^98 + 15 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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